Answer :
To determine if [tex]\(x - 3\)[/tex] is a factor of [tex]\(x^3 - 3x^2 - 6x + 36\)[/tex], we need to perform polynomial division to find the remainder. If the remainder is zero, then [tex]\(x - 3\)[/tex] is a factor. If the remainder is not zero, then [tex]\(x - 3\)[/tex] is not a factor.
Here's a step-by-step breakdown of the process:
1. Set Up the Division: We are dividing the polynomial [tex]\(x^3 - 3x^2 - 6x + 36\)[/tex] by [tex]\(x - 3\)[/tex].
2. Divide the First Terms:
- Divide the leading term of the dividend [tex]\(x^3\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex], which gives [tex]\(x^2\)[/tex].
- Multiply [tex]\(x^2\)[/tex] by the entire divisor [tex]\(x - 3\)[/tex] which results in [tex]\(x^3 - 3x^2\)[/tex].
3. Subtract:
- Subtract [tex]\(x^3 - 3x^2\)[/tex] from [tex]\(x^3 - 3x^2 - 6x + 36\)[/tex], which results in [tex]\(0x^3 + 0x^2 - 6x + 36\)[/tex] or simply [tex]\(-6x + 36\)[/tex].
4. Repeat the Process:
- Divide the new leading term [tex]\(-6x\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(-6\)[/tex].
- Multiply [tex]\(-6\)[/tex] by [tex]\(x - 3\)[/tex] which gives [tex]\(-6x + 18\)[/tex].
- Subtract [tex]\(-6x + 18\)[/tex] from [tex]\(-6x + 36\)[/tex] to get [tex]\(18\)[/tex].
5. Check the Remainder:
- The remainder after performing the division is [tex]\(18\)[/tex].
Since the remainder is [tex]\(18\)[/tex], which is not equal to zero, [tex]\(x - 3\)[/tex] is not a factor of [tex]\(x^3 - 3x^2 - 6x + 36\)[/tex]. Therefore, the remainder when you divide is 18.
Here's a step-by-step breakdown of the process:
1. Set Up the Division: We are dividing the polynomial [tex]\(x^3 - 3x^2 - 6x + 36\)[/tex] by [tex]\(x - 3\)[/tex].
2. Divide the First Terms:
- Divide the leading term of the dividend [tex]\(x^3\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex], which gives [tex]\(x^2\)[/tex].
- Multiply [tex]\(x^2\)[/tex] by the entire divisor [tex]\(x - 3\)[/tex] which results in [tex]\(x^3 - 3x^2\)[/tex].
3. Subtract:
- Subtract [tex]\(x^3 - 3x^2\)[/tex] from [tex]\(x^3 - 3x^2 - 6x + 36\)[/tex], which results in [tex]\(0x^3 + 0x^2 - 6x + 36\)[/tex] or simply [tex]\(-6x + 36\)[/tex].
4. Repeat the Process:
- Divide the new leading term [tex]\(-6x\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(-6\)[/tex].
- Multiply [tex]\(-6\)[/tex] by [tex]\(x - 3\)[/tex] which gives [tex]\(-6x + 18\)[/tex].
- Subtract [tex]\(-6x + 18\)[/tex] from [tex]\(-6x + 36\)[/tex] to get [tex]\(18\)[/tex].
5. Check the Remainder:
- The remainder after performing the division is [tex]\(18\)[/tex].
Since the remainder is [tex]\(18\)[/tex], which is not equal to zero, [tex]\(x - 3\)[/tex] is not a factor of [tex]\(x^3 - 3x^2 - 6x + 36\)[/tex]. Therefore, the remainder when you divide is 18.
